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Introduction
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Mathematics
Mathematical Analysis

A variation norm Carleson theorem

Profile image of Oguzhan OzenOguzhan Ozen

2009, arXiv (Cornell University)

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83 pages

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Abstract

We strengthen the Carleson-Hunt theorem by proving L p estimates for the r-variation of the partial sum operators for Fourier series and integrals, for p > max{r ′ , 2}. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

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  37. James T. Campbell, Roger L. Jones, Karin Reinhold and Máté Wierdl, Oscillation and variation for the Hilbert transform, Duke Math. J., 105 (2000), 59-83.
  38. Alberto P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968) 349-353.
  39. Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157.
  40. Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409-425.
  41. WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal. 179 (2001), no. 2, 426-447.
  42. Ronald R. Coifman and Guido Weiss, Transference methods in analysis. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31. American Mathematical Society, Providence, R.I., 1976.
  43. Ciprian Demeter, Michael T. Lacey, Terence Tao, and Christoph Thiele, Breaking the duality in the return times theorem, Duke Math. J. 143 (2008), no. 2, 281-355.
  44. The Walsh model for M * 2 Carleson, Rev. Mat. Iberoamericana 24 (2008), no. 3, 721-744.
  45. Charles Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551-571.
  46. Charles Fefferman and Elias M. Stein, H p spaces of several variables. Acta Math. 129 (1972), no. 3-4, 137-193.
  47. Loukas Grafakos, Terence Tao, and Erin Terwilleger, L p bounds for a maximal dyadic sum operator, Math. Z. 246 (2004), no. 1-2, 321-337.
  48. Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235-255.
  49. Roger L. Jones, Robert Kaufman, Joseph Rosenblatt, and Máté Wierdl, Oscillation in ergodic theory, Erg. Th. & Dyn. Sys., 18 (1998), 889-936.
  50. Roger L. Jones, Andreas Seeger, and James Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6711-6742.
  51. Roger L. Jones and Gang Wang, Variation inequalities for the Fejér and Poisson kernels, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4493-4518 .
  52. Carlos E. Kenig and Peter A. Tomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), no. 1, 79-83.
  53. Michael Lacey, Issues related to Rubio de Francia's Littlewood-Paley inequality, NYJM Monographs, 2. State University of New York, University at Albany, Albany, NY, (2007) 36 pp. (electronic).
  54. Michael Lacey and Erin Terwilleger, A Wiener-Wintner theorem for the Hilbert transform, Ark. Mat. 46 (2008), no. 2, 315-336.
  55. Michael Lacey and Christoph Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7 (2000), no. 4, 361-370.
  56. Karel de Leeuw, On Lp multipliers, Ann. of Math. 81 (1965), 364-379.
  57. Dominique Lépingle, La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 4, 295-316.
  58. Terry Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett. 1 (1994), no. 4, 451-464.
  59. Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), 215- 310.
  60. Camil Muscalu, Terence Tao, and Christoph Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity 16 (2003), no. 1, 219-246.
  61. A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett. 10 (2003) no. 2-3, 237-246.
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  65. Tong Seng Quek, Littlewood-Paley type inequality on R, Math. Nachr. 248/249 (2003), 151-157.
  66. José L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana 1 (1985), no. 2, 1-14.
  67. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971.
  68. Terence Tao and Christoph Thiele, Nonlinear Fourier analysis, IAS/Park City Math. Ser., to appear.
  69. Christoph Thiele, Wave packet analysis, CBMS Regional Conference Series in Mathematics, vol. 105, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006.
  70. Norbert Wiener, The quadratic variation of a function and its Fourier coefficients, MIT Journal of Math. and Physics 3(1924) pp. 72-94.
  71. R. Oberlin, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

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